Unimodular homotopy algebras and Chern–Simons theory

Quantum Chern–Simons invariants of differentiable manifolds are analyzed from the point of view of homological algebra. Given a manifold M   and a Lie (or, more generally, an L∞L∞) algebra gg, the vector space H⁎(M)⊗gH⁎(M)⊗g has the structure of an L∞L∞ algebra whose homotopy type is a homotopy invariant of M  . We formulate necessary and sufficient conditions for this L∞L∞ algebra to have a quantum lift. We also obtain structural results on unimodular L∞L∞ algebras and introduce a doubling construction which links unimodular and cyclic L∞L∞ algebras.